Economic modelling using computational intelligence techniques
(2013)

Attempting to successfully and accurately predict the financial market has long attracted the interests and attention of economists, bankers, mathematicians and scientists alike. The financial markets form the bedrock of any economy. There are a large number of factors and parameters that influence the direction, volume, price and flow of traded stocks. This coupled with the markets’ vulnerability to external and non-finance related factors and the resulting intrinsic volatility makes the development of a robust and accurate financial market prediction model an interesting research and engineering problem. In an attempt to solve this engineering problem, the authors of this paper present a rough set theory based predictive model for the financial markets. Rough set theory has, as its base, imperfect data analysis and approximation. The theory is used to extract a set of reducts and a set of trading rules based on trading data of the Johannesburg Stock Exchange (JSE) for the period 1 April 2006 to 1 April 2011. To increase the efficiency of the model four of dicretization algorithms were used on the data set, namely (Equal Frequency Binning (EFB), Boolean Reasoning, Entropy and the Naïve Algorithm. The EFB algorithm gives the least number of rules and highest accuracy. Next, the reducts are extracted using the Genetic Algorithm and finally the set of dependency rules are generated from the set of reducts. A rough set confusion matrix is used to assess the accuracy of the model. The model gave a prediction accuracy of 80.4% using the Standard Voting classifier. Keywords-rough set theory; financial market modelling, neural networks, discretization, classification. INTRODUCTION Around the world, trading in the stock market has gained enormous popularity as a means through which one can reap huge profits. Attempting to successfully and accurately predict the financial market has long attracted the interests and attention of economists, bankers, mathematicians and scientists alike. Thus, stock price movement prediction has long been a cherished desire of investors, speculators and industries [1]. For a long time statistical techniques such as Bayesian models, regression and some econometric techniques have dominated research activities in prediction [2]. The primary approach to financial forecasting has been the identification of a stock price trend and continuation of the investment strategy until evidence suggests that the trend has reversed [3]. One of the biggest problems with the use of regression methods is that they fail to give satisfactory forecasting results for some series’ because of their linear structure and other inherent limitations [8, 9]. However the emergence of computational intelligence techniques as a viable alternative to the “traditional” statistical models that have dominated this area since the 1930’s [2,3] has given impetus to the increasing usage of these techniques in fields such as economics and finance [3]. Apart from these, there have been many other successful applications of intelligent systems to decision support and complex automation tasks [4 6]. Since the year of its inception in 1982, rough set theory has been extensively used as an effective data mining and knowledge discovery technique in numerous applications in the finance, investment and banking fields [4, 7, 11, 12, 20]. Data mining is a discipline in computational intelligence that deals with knowledge discovery, data analysis, and full and semi-autonomous decision making [7]. It entails the analysis of data sets such that unsuspected relationships among data objects are found. Predictive modeling is the practice of deriving future inferences based on these relationships. The financial markets form the bedrock of any economy. There are a large number of factors and parameters that influence the direction, volume, price and flow of traded stocks. This coupled with the markets’ vulnerability to external and nonfinance related factors and the resulting intrinsic volatility makes the development of a robust and accurate financial market prediction model an interesting research and engineering problem. This paper presents a generic stock price prediction model based on rough set theory. The model is derived on data from the daily movements of the Johannesburg Stock Exchange’s All Share Index. The data used was collected over a five year period from 1 April 2006 to 1 April 2011. The methodology used in this paper is as follows: data preprocessing, data splitting, data discreetization, redundant attribute elimination, reduct generation, rule generation and prediction. 71 The rest of this paper is organized as follows: Section II covers the theoretical foundations of rough set theory, Section III gives the design of the proposed prediction model while an analysis of the results and conclusions are presented in Sections IV and V respectively. THEORETICAL FOUNDATIONS OF ROUGH SETS Rough set theory (RST) was introduced by Pawlak in 1982. The theory can be regarded as a mathematical tool used for imperfect data analysis [10]. Thus RST has proved useful in applications spanning the engineering, financial and decision support domains to mention but a few. It is based on the assumption that “with every object in the universe of discourse, some information (data or knowledge) is associated” [10]. In practical applications, the “universe of discourse” described in [10] is usually a table called the decision table in which the rows are objects or data elements and the columns are attributes and the entries are called the attribute values [3]. The objects or data elements described by the same attributes are said to be indiscernible (indistinguishable) by the attribute set. Any set of indiscernible data elements forms a granule or atom of knowledge about the entire “universe of discourse” (information system framework) [10]. A union of these elementary sets (granules) is said to be a precise or crisp set, other-wise the set is said to be rough [1,2,7,10]. Every rough set will have boundary cases i.e data objects which cannot certainly be classified as belonging to the set or its complement when using the available information [10]. Associated with every rough set is a pair of sets called the lower and upper approximation of the rough set. The lower approximation consists of those objects which one can definitively say belong to the target set. The upper approximation consists of those objects which possibly belong to the target set. The difference between the two sets is the boundary region. The decision rule derived specifies an outcome based on certain conditions. Where the derived rule uniquely identifies outcomes based on some conditions the rule is said to be certain else it is uncertain. Every decision rule has a pair of probabilities associated with it, the certainty and coverage coefficients [3]. These conditional probabilities also satisfy Bayes’ theorem [7,10]. The certainty coefficient is the conditional probability that an object that belongs to the decision class outlined by the rule given that it satisfies the conditions of the rule. The coverage coefficient on other hand expresses the conditional probability of reasons given some decision [10]. Clearly RST can be seen to overlap with many other theories in the realm of imperfect knowledge analysis such as evidence theory, Bayesian inference, fuzzy sets etc [1, 3, 4, 10, 11, 12]. To define rough sets mathematically, we begin by defining an information system S = (U,A), where U and A are finite and non-empty sets that represent the data objects and attributes respectively. Every attribute has a set of possible values Va. Va is called the domain of a. A subset of A say B will determine a binary relation I(B) on U, which is called the indiscerniblity relation. The relation is defined as follows: ( ) ( ) if and only if a(x) = a(y) for every a in B, where a(x) denotes the value of attribute a for data object x [10]. I(B) is an equivalence relation. All equivalence classes of I(B) as U/I(B). An equivalence class of I(B) containing x is denoted as B(x). If (x,y) belong to I(B) they are said to be indiscernible with respect to B. All equivalence classes of the indescernibility relation, I(B), are referred to as B-granules or B-elementary sets [10]. In the information system defined above, we define as in [10]: (1) And, (2) We now define the two operators assigned to every (1) two sets called the upper and lower approximation of X. The two sets are defined as follows [10]: ( ) ⋃ { ( ) ( ) } (3) And, ( ) ⋃ { ( ) ( ) } (4) Thus, the lower approximation is the union of all Belementary sets that are included in the target set, whilst the upper approximation is the union of all B-elementary sets that have a non-empty intersection with the target set. The difference between the two sets is called the boundary of region of X.